Bicontinuous minimal surface nanostructures for polymer blend solar cells

KIMBER, R. G. E., WALKER, A. B., SCHRODER-TURK, G. E. and CLEAVER, D. J. <http://orcid.org/0000-0002-4278-0098>

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KIMBER, R. G. E., WALKER, A. B., SCHRODER-TURK, G. E. and CLEAVER, D. J. (2010). Bicontinuous minimal surface nanostructures for polymer blend solar cells. Phys. Chem. Chem. Phys., 12 (4), 844-851.

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Bicontinuous minimal surface nanostructures for

polymer blend solar cells

Robin G. E. Kimber∗ Alison B. Walker†

Gerd E. Schröder-Turk‡ Douglas J. Cleaver§

October 2, 2009

Abstract

This paper presents the first examination of the potential for bicontinuous struc-

tures such as the gyroid structure to produce high efficiencysolar cells based on

conjugated polymers. The solar cell characteristics are predicted by a simulation

model that shows how the morphology influences device performance through in-

tegration of all the processes occurring in organic photocells in a specifed mor-

phology. In bicontinuous phases, the surface defining the interface between the

electron and hole transporting phases divides the volume into two disjoint subvol-

umes. Exciton loss is reduced because the interface at whichcharge separation

occurs permeates the device so excitons have only a short distance to reach the

interface. As each of the component phases is connected, charges will be able to

reach the electrodes more easily. In simulations of the current-voltage character-

istics of organic cells with gyroid, disordered blend and vertical rod (rods normal

to the electrodes) morphologies, we find that gyroids have a lower than anticipated

performance advantage over disordered blends, and that vertical rods are superior.

These results are explored thoroughly, with geminate recombination, i.e. recom-

bination of charges originating from the same exciton, identified as the primary

∗Department of Physics, University of Bath, Bath, BA2 7AY, UK†Department of Physics, University of Bath, Bath, BA2 7AY, UKEmail: a.b.walker�bath.a .uk‡Institut für Theoretische Physik, Friedrich-Alexander Universität, Erlangen-Nürnberg, Staudtstraße 7,

D-91058 Erlangen, Germany§Materials and Engineering Research Institute, Sheffield Hallam University, Howard Street, Sheffield, S1

1WB, UK

1

source of loss. Thus, if an appropriate materials choice could reduce geminate

recombination, gyroids show great promise for future research and applications.

Introduction

Organic photovoltaics have the potential to produce cheap solar power in lightweight,

flexible and portable cells.1,2 Compared to their inorganic counterparts, organic cells

are a relatively new technology, with several issues still to be addressed before be-

coming commercially viable. There are several types of organic cells: hybrid or-

ganic/inorganic cells,3 and all-organic cells such as conjugated polymer-fullerene com-

posites with power conversion efficiencies exceeding 5%,4–6 and conjugated polymer

blend cells with efficiencies as high as 1.8%.7,8

In polymer blend solar cells, absorbed photons generate excitons, which dissociate

into electron-hole pairs at the interface between the two polymers, which are then ex-

tracted by the electrodes to create a photocurrent. The morphology of the active layer

of organic photovoltaics at the nanoscale is of great interest, as it impacts strongly

on device performance.2 Due to the low exciton diffusion length, around 10-20 nm,2

compared to the optical absorption length, typically 100 nm, a bulk heterojunction

morphology that employs an interpenetrating network of thecomponent polymers is

frequently employed. A finely intermixed morphology with small feature sizes is best

for exciton dissociation, however, a coarse morphology is required for collection of the

photogenerated charges. There is therefore a tradeoff between these two processes as

has been demonstrated by Dynamical Monte Carlo (DMC) simulations that we adapted

from surface physics to look at charge and energetic processes in any morphology on

the nanometre scale.9 Since our original work, DMC has been used as a general inves-

tigation of morphology,10–12or to examine specific loss mechanisms.13,14

Drift-diffusion modelling has also been successfully usedto model organic so-

lar cells.15–17 However, such models are unable to take full account of the effect

of three-dimensional structures, which are of interest here, instead reducing the

complex morphology to a homogeneous system or a simplified two-dimensional

2

structure. These models, apart from,17 also do not normally take full account

of the related process of exciton dissociation, which has multiple stages that are

critically influenced by morphology.13,18

The blend structures used in solar cells are in general highly disordered and are

prone to the development of disconnected islands which limit charge collection. Thus,

the identification of routes to more efficient structures, aswell as a greater understand-

ing of loss mechanisms, is highly desirable. Self-assembling ordered microphase-

separated geometries such as the gyroid morphology can be seen in diblock copoly-

mers19 and certain dendrimer systems.20 In principle, such systems should yield im-

proved phase connectivity when compared with the basic percolation pathways offered

by random blends. The observation that such phases can freely self-assemble from

systems as simple as appropriately-shaped hard particles21 indicates that there is no

length-scale limitation on the periodicities of these phases. We speculate, therefore,

that there are no fundamental materials problems that wouldprevent realisation of a

hybrid solar cell device based on a gyroid morphology.

These bicontinuous morphologies have a large interfacial area and continuous trans-

port pathways, and the generation of morphologies by self-assembly makes such struc-

tures highly reproducible.22 Gyroid morphologies have been created in nanoporous

films by, amongst others, Urade et al.23 Bulk heterojunction solar cells made from

a porous titania structure synthesised using a poly(styrene-block-polyethylene oxide)

diblock copolymer template infiltrated with a semiconducting polymer were made by

Oey et al,24 who noted that gyroid structures could be created by this technique. Re-

cently, gyroid and columnar structures were replicated in anatase titania and the ti-

tania structures immersed in dye and back filled with a liquidelectrolyte to create

dye-sensitised cells (DSCs).25,26 These cells show power efficiencies of around 2%,

considerably less than the best efficiencies of 11% obtainedfor liquid electrolyte dye-

sensitised cells made from mesoporous titania,27 but comparable to devices of a sim-

ilar thickness. To the authors’ knowledge,such bicontinuous structureshave yet to

be adopted in an all-conjugated polymer solar cell, although diblock copolymers based

on conjugated polymers have now been synthesised.28

3

Here, we present the results of DMC simulations of polymer solar cells employ-

ing a range of morphology classes. We examine three bicontinuous morphologies, as

formed in diblock copolymers: the gyroid, double gyroid anddouble diamond phases.

We assess the feasibility of these structures on different length scales as novel active

layer morphologies, in comparison to two other classes: thedisordered blend and in-

terdigitated vertical rod morphologies. The interdigitated structure used is similar to

the cylindrical phase, which can also form in such materials.29 These structures are

illustrated in 1. To date, nearly all the reported DMC simulations have only looked at

short-circuit, and been limited to a small range of morphologies. Here, we extend DMC

to open-circuit, and separately examine geminate and bimolecular recombination, and

the impact of high illumination.

The polymer backbone in conjugated polymers may be too rigidfor a gyroid

phase to form, and to date it is yet to be observed.30,31 S. Sun has suggested the

use of a flexible non-conjugated bridge unit in order to overcome this problem.32

However, our emphasis here is on the usefulness of bicontinuous structures due

to the recent interest in such structures, and the possibility that these structures

could be made in the near future. All three structures studied here share the

essential traits of island-free continuous charge transport pathways with a high

interfacial area, which should provide a clear performanceimprovement on exist-

ing disordered blend structures.

Model

Within each morphology class, a range of morphologies with different feature sizes

have been created. The feature sizel f = 3rVS, whererVS is the volume-to-surface

ratio,10 provides a rough approximation for the average domain sizes. While more

direct measures for domain size are available,33–36 the method used here is sufficient

to identify performance trends. The solar cell is modelled by 64× 64× 64 voxels each

of dimensions 1 nm3 where each voxel represents a hopping site. Periodic boundary

conditions are applied in they andz dimensions, with electrodes atx = 0 andx = 64

4

nm, parallel to they-z plane. Voxels are labelled as either acceptor (n) or donor (p),

with a 1:1 ratio in every morphology.

Disordered blend morphologies with increasing feature size were created from the

Ising model.9 Interdigitated rod structures of different widths were produced in a chess-

board layout. Triply-periodic bicontinuous morphologieswere derived from the gyroid

and diamond triply-periodic minimal surfaces of cubic symmetry.37,38 These surfaces

have vanishing mean curvature, are periodic in three perpendicular directions and di-

vide space into two identical labyrinth-like domains each of which represents a con-

tinuous tunnel network extending periodically throughoutspace. The gyroid and the

diamond surfaces are examples of triply-periodic minimal surfaces that are structural

models for certain mesophases in copolymer blends and surfactant/lipid mixtures.39 A

full descriptions of how these morphologies have been created has been included in the

supporting information. Scaled down versions of each structure were used as a unit cell

for larger, multiply-connected structures to creating morphologies of the same volume

but varyingl f , 1, panel (f). For comparison, a bilayer structure is also studied.

Excitons enter the system at a rate determined by the absorption profile and execute

a random walk in the phase in which they are created at a hopping rate determined by

Förster theory40

wi j = wex

(

R0

Ri j

)6

f (Ei ,E j) (1)

whereRi j is the distance between hopping sitesi and j andR0 is the Förster radius.Ei

andE j are the energies of sitesi and j. The attempt to hop frequencyweR60 represents

an average of inter- and intra-chain hopping.We do not distinguish between inter-

and intra-chain exciton or charge motion as it is hard to ascertain how the chains

pack together. Not accounting for the full chain picture also saves considerably on

computing time whilst still allowing us to examine the parameters of interest.

The factor

f (Ei ,E j) = exp

(

−

E j −Ei

kBT

)

E j > Ei , f (Ei ,E j) = 1 E j < Ei . (2)

describes the influence of site energies on hopping rate.Ei andE j are determined from

5

Figure 1: Disordered blend (a) and interdigitated rod (b) structures. Panels (c-e) showa single cubic translational unit cell of the single gyroid with symmetryI4132 (c), theIa3d double gyroid (d) and thePn3m double diamond (e) structures. Panel (f) showsan extended gyroid structure consisting of 23 translational unit cells of the structure in(c) with half the lattice lengtha.

a Gaussian distributed density of states (DOS) of widthσ , which describes material

disorder. For simplicity, the same density of states is usedfor both charges and exci-

tons. Instead of hopping, the exciton may recombine at a rateof τ−1ex .

If an exciton encounters an interface between then- andp-type polymer, a charge

transfer state (bound charge pair) may be created at a rate ofkdiss. This state can form

a geminate electron-hole pair or recombine at a ratekr , at which point it is considered

6

to be lost from the system.13

The charges created by exciton dissociation or by injectionare allowed to hop from

sitesi to j where sitej is a nearest neighbour of the same polymer as sitei at a rate

obtained from Marcus theory,41

Ri j = νhop exp

[

−

(E j −Ei + λ )2

4λkBT

]

. (3)

Here,λ corresponds to the chain reorganisation energy and the prefactorνhop encom-

passes the electronic coupling between states and the distance between hopping sites,

both of which are taken as constant for nearest-neighbour hopping in a single system.

Theith site is initially assigned an energy

Ei = Eσ i + φ + ∆in j −eFxi −e2

16πε0εrxi(4)

whereEσ i is chosen from a Gaussian distribution as noted above,φ is the work function

of the Al contact,∆in j is the injection barrier,F is the net field resultant from built-in

voltage (Vbi) and the applied bias,xi is the distance from the contact, and the final term

describes image charge effects for a material of dielectricconstantε0εr whereε0 is the

vacuum permittivity ande is the electron charge magnitude. An equivalent expression

exists for hole transport.

When calculating hopping rates for charge carriers,Ei and E j in Eq. 3 are

further updated by a factor △ E to take account of the coulombic effect of all

other charges within the system, where

△ E =n

∑j=1

qe4πε0εr r i j

(5)

where n+ 1 is the total number of charges in the system,q = +e for electron-

electron and hole-hole repulsion and−e for electron-hole attraction and r i j is the

distance between chargesi and j. At the charge densities seen here this approach

is computationally far more efficient than updating all the sites within a speci-

fied cutoff radius of the moving charge carrier, and the number of calculations

7

scales withn, the number of charges. Although coulombic interactions are up-

dated when calculating each new event, waiting times for events already held in

the queue are not. This procedure saves computing time without compromising

on simulation accuracy42

Once a hop has occurred, the possibilities are: further hopping, geminate or bi-

molecular recombination, or extraction at an electrode; depending on the location of

the charge within the morphology and with respect to other particles. Charges may also

be injected, via a hop from the Fermi level of the electrode over the injection barrier

into the first monolayer.42,43

Simulation of charge and exciton dynamics is carried out using the First Reaction

Method (FRM) algorithm. Each event has an associated waiting timeτ = (1/w) ln(X)

based on its ratew and a random numberX uniformly distributed between 0 and 1.

The events are stored in a queue in order of increasingτ. At each timestep, the event

at the start of the queue is executed, and then removed. The waiting times of all events

remaining in the queue are then reduced by the time expired. Site energies are updated,

and new events enabled and existing ones disabled as appropriate. To reduce com-

puting resource, events which are already stored in the queue are based on an earlier

state of the system even though the system may have changed significantly, an ap-

proximation validated by.42 Simulations are allowed to continue until steady-state has

been maintained for enough time to build up useful data, and are repeated to reduce

uncertainty.

To characterise the morphologies at a detailed level we calculate the fraction of ex-

citons which dissociateηed; the fraction of charges that recombine with their geminate

twins ηgr; that recombine with other charges (bimolecular recombination) ηbr; and

that are extracted, the charge collection efficiencyηcc. The internal quantum efficiency

(IQE) is the productηcc×ηed. The power conversion efficiency (PCE), is the ratio of

the power per unit area at the maximum power operating point delivered by the device,

obtained from the current densityJm and voltageVm at that point, to the incident power

8

per unit area at AM 1.5 illumination. The fill factor

FF =JmVm

JSCVOC(6)

whereJSC is the short-circuit current density, andVOC is the open-circuit voltage.

As far as possible, we take our system parameters from experimental studies of the

hole transporting copolymer poly(9,9’-dioctylfluorene-co-bis(N,N’-(4,butylphenyl))bis(N,N’-

phenyl-1,4-phenylene)diamine) (PFB) blended with the electron transporting copoly-

mer poly(9,9’-dioctylfluorene-co-benzothiadiazole) (F8BT), where the hole extracting

electrode is ITO/PEDOT:PSS where ITO is Indium Tin Oxide andPEDOT:PSS is

Poly(3,4-ethylenedioxythiophene) poly(styrenesulfonate) and the electron extracting

electrode is Al, as this system has been extensively studied.10,44,45From spectral data,

this blend has been calculated to absorb 5.8% of the AM (air mass) 1.5 solar emission

spectrum, with an assumed Gaussian optical field profile.9 We setwexR60 = 0.02, such

that simulations of excitons in a single component materialwith energetic disorderσ

= 0.062 eV gives a diffusion lengthLex of 6 nm46 in a lifetimeτex of 500 ps.47 Taking

λ = 0.75 eV and a mobility ofµ0 = 1× 10−8 m2Vs−1 for both polymers,48,49 and by

assuming isoenergetic material, we calculateνhop = 2.41 ps−1 using the Einstein rela-

tionship. Other parameter values arekdiss = (100 fs)−1,50 kr = 1 × 106 s−1, ε = 4,Vbi

= 1.3 V. ∆in j = 0.8 eV for electrons and 0.1 eV for holes based on the HOMO/LUMO

levels of the polymers and the work functions of the electrodes.51–53The uncertainty

in the hole injection barrier height was overcome by fitting dark current mod-

elling to published experimental results,45 ensuring the charge density and local

field near the electrodes are correct.

Results and discussion

2 (a) compares the IQE of the five different morphology classes, as a function of feature

size. Given the disparity in feature sizes simulated, interpolation has been used to find

the peak efficiency of each morphology class shown in 1. All the morphology classes

9

(a) (b)

(c)

Figure 2: a) IQE, b) FF and c) PCE as a function of feature size for blends (solid line,▽), rods (solid line,N), gyroids (dashed line,•), double gyroid (dashed line,�) anddouble diamond (dashed line,�).

reproduced the well-established behaviour of an idealisedintermediate morphology,

where the product ofηed andηcc is optimised. Morphology class has little effect on

the optimum feature size, which is close to the exciton diffusion length (6 nm) in each

case. Overall, the gyroid, double gyroid and double diamondstructures do not com-

pare as favourably with the disordered blend morphologies as might be expected. At

the peak IQE, the bicontinuous structures are fractionallymore efficient than the blend.

These results are sensitive to device parameters,9 many of which are only known ap-

proximately as they depend on chain alignments, for examplethe dissociation rates,54

exciton diffusion coefficient47 and mobilities,55 and these alignments will vary widely

in the disordered films. Another problem is domain purity as minority components in

10

Table 1: Interpolated peak values of IQE and PCE, and the feature size at which theyoccur. Fill factor values are taken at the peak PCE.

IQE PCE FFBlend 0.55 (5.6 nm) 0.56% (7.5 nm) 0.34Rod 0.77 (6.5 nm) 0.91% (8.0 nm) 0.38Gyroid 0.57 (5.6 nm) 0.53% (8.6 nm) 0.33Double gyroid 0.58 (6.8 nm) 0.57% (8.3 nm) 0.32Double diamond 0.59 (6.4 nm) 0.56% (6.9 nm) 0.30

the blend domains can have a measurable impact on the resultsas they act as dissoci-

ation, and hence recombination, centres. Recent experimental studies have found that

the domain size of an optimised morphology can exceed the exciton diffusion length45

even though domain purity >90% is common.56,57We repeated the calculations of IQE

in the blend morphologies, this time creating additional islands by swopping voxels in

the domains of each polymer, and then plotted the results against the original feature

sizes to mimic experiment where the islands are invisible under AFM. The peak IQE is

reduced from 0.55 at a feature size of 5.6 nm to 0.50 at a feature size of 6.4 nm if 1%

impurities are introduced and to 0.48 at a feature size of 7.5nm if there are 5% impu-

rities. The additional islands reduce the drop-off in exciton dissociation efficiency as

l f increases. However,ηcc no longer increases monotonically with feature size across

the range of morphologies, but shows a peak because exciton dissociation preferen-

tially takes place at the islands as the interfacial area of the charge transport pathways

decreases, and such charges are unable to escape and contribute to the current. At the

peak IQE, most dissociation takes place at connected features, and so the peak IQE

value drops only marginally. Self-assembled nanostructures are less likely to contain

these islands, enhancing their appeal.

From 2 (b), FF can be seen to increase steadily withl f , in agreement with experi-

ment,45,58,59due to the morphological dependence ofηcc. In the blends, FF is≈0.25

at the smallest feature sizes modelled, suggesting drift-only transport which produces

a linear J-V curve. The features in these morphologies are sosmall that charges follow

extremely tortuous paths. As there are interfaces close to each other throughout the

device, charge generation is uniformly spread so there are only small charge gradients,

11

minimising the diffusion current. The range of FF values is similar to experimental

findings45 and for the limiting case of a bilayer structure (not shown) FF is 0.54.

2 (c) shows that PCEs are comparable for the novel structuresand the blends. The

feature size of the optimum morphology increases noticeably when examining com-

plete J-V performance, compared to characterisation in terms of IQE alone, due to the

morphological dependence of FF. Thus a morphology optimised at short circuit will

not be the optimal morphology for power conversion. Vertical rod morphologies are

consistently superior to all others examined. The lower than expected IQE values in the

bicontinuous morphologies can be understood by examining the recombination data in

3.

Increasingl f may be expected to reduce geminate recombination because the charges

have more room in which to escape their geminate twin. For blend morphologies, the

large drop inηgr(l f ) up till 3 nm is attributed to the difficulty for an initially separated

geminate pair at the smallest separations to escape their mutual coulombic well. At

l f =3 nm there is a sharp change in the drop off rate forηgr(l f ). For l f > 3nm, charges

can avoid recombination, butηgr(l f ) decreases further because the blend structure mor-

phology can force charges back together en route to the electrodes. Asl f continues to

increase, the charges have more room in which to escape theirgeminate twin soηgr

is less sensitive tol f . Such behaviour was seen and explained by Groves et. al.13

for bilayer and blend devices. Single and double gyroid structures, however, exhibit

an increase inηgr with l f for l f > 3nm, an effect which will be discussed below with

reference to illumination level.

It is not possible to create the single gyroid and double diamond morphologies for

l f under 2 nm, but for the other morphologies,ηbr takes its lowest values at these

small feature sizes, as can be seen in 3(b). This result is a consequence ofηgr being

large for these morphologies, removing charges that could otherwise suffer bimolecular

recombination. Onceηgr reaches a low value asl f approaches 3 nm, there is a sharp

increase inηbr. For l f > 3 nmηbr drops off for all morphologies, although the single

gyroid shows slightly less bimolecular recombination thanthe blends at larger feature

sizes. Increasingl f reduces the probability of an exciton reaching an interfaceand

12

dissociating, lowering the charge density. It also increases the average distance between

domains, so that dissociated charges are less likely to encounter other charges of the

opposite type after escaping their geminate twins. In14 ηbr was predicted for different

feature sizes with related conclusions.

In general the novel morphologies appear to be no better thanblends at either

achieving separation of the geminate pair, or preventing recombination en route to the

electrodes. We infer that the presence of islands as well as discontinuous and ’cul-de-

sac’ pathways in the blends is not a severe limitation in their efficiency. Rod structures

exhibit a much lower level of geminate recombination at almost every feature size.

This advantage is lost at the two smallest rod sizes, 1 nm and 2nm, because, as for the

smallest blend structures, motion perpendicular to the interface is tightly constrained

so separation is highly unlikely, as charges are unable to escape their mutual coulombic

well. In this case interfacial tracking is more likely to occur, increasing the number of

recombination attempts. However, the short pathways for extraction in the rod struc-

tures ensures bimolecular recombination is minimal even for the narrowest rods, where

charges created within different rods can also remain entirely isolated from one an-

other. Furthermore, unlike the novel structures, charges in the rod structures only move

parallel to the field with a direct route to the electrodes, reducing escape time.

To further characterise the bicontinuous morphologies with respect to the other

structures, the same simulations were performed at 5 suns illumination, to examine the

influence of high charge densities on performance. The polymers simulated here have

low absorption coefficients, so the charge densities at thismuch higher illumination

level show effects that may be seen in other materials at 1 sun. The IQEs for the

different morphology classes and the geminate and bimolecular recombination levels

are given in Figures 1-3 of the supporting information, and can be compared to 2 (a)

and 3(a) and (b). We find that geminate recombination is decreased by up to 25% for

the blends and novel structures, but increases fractionally for the rod structures. We

attribute this decrease to diffusion away from curved surfaces, which cannot occur in

the rod structures. However, the decreased geminate recombination does not result

in an increased FF or IQE, asηbr increases by a factor of 2 to 3, making it a loss

13

process now comparable to the geminate mechanism. This result leads to a decrease

in the efficiency of the blends and bicontinuous structures by 12-25%, but the rods by

only 8%, widening their advantage over the other structures. In the blends, the peak

IQE shifts from the 5.6 nm to 8.6 nm morphology as wider pathways are required

to keep recombination at a reasonable level. We can conclude, then, that the novel

structures are no better at handling higher charge densities than the blends. The level

of bimolecular recombination increases by the same amount,further evidence that the

disordered structure of the blends is not primarily responsible for their inefficiency.

As already mentioned, at normal illumination the gyroid anddouble gyroid struc-

tures exhibit an increase inηgr with l f , a trend which we now see in the blend structures

at 5 suns illumination whenl f >≈ 10 nm. We speculate that this increase occurs be-

causethe effect on the local field ofthe presence of multiple charge pairs along the

same planar stretch of interface interferes with charge separation, an effect which oc-

curs more strongly for larger domains. The existence of thiseffect is confirmed by

intensity dependent simulations up to 10 suns (not shown) ofthe bilayer and rod struc-

tures, where all the surfaces are planar. We find thatηgr increases with illumination,

though more weakly for the rods due to the greater surface area. We have already seen

that, at 1 sun illumination, this effect is absent in the blends but present in the gyroid

and double gyroid structures. Additional simulations showthe effect is absent in the

latter structures at 0.01 suns, so we deduce that the influence of neighbouring geminate

pairs on the attraction felt by charges within a geminate pair becomes important for the

single and double gyroid structures at a lower illuminationlevel than the blends.

The charge densities simulated here are of the order of 1021 - 1022 m−3, consis-

tent with other models,17,44,60which in a system of this size consists of no more

than approximately 10 charge pairs at any time. Animations show that these

charge pairs remain in the viscinity of each other for a long time after dissociation,

and can track each other within the device. The reason the novel morphologies are

more influenced by this effect is due to the continuous natureof their interface,

which allows charge pairs to track each other throughout thedevice, whereas the

blends have broken interfaces which reduce tracking.This effect competes with the

14

two already described: difficulty in escaping the mutual coulombic well at very small

feature sizes, and diffusion from curved surfaces.

It is surprising that rods perform so much better than blends, whereas the novel

morphologies are, at best, only marginally better. Rods have the shortest charge trans-

port pathways, and at small feature sizes can keep charges entirely isolated from each

other. However, this effect does not sufficiently explain the large difference in geminate

recombination levels. In the simulation, the probability of separation following disso-

ciation is determined by the competition between the field dependent rate for one of

the charges to hop away from the interface and the constant recombination ratekr . Eq.

3 shows that parallel and anti-parallel hops with respect tothe field result in equal and

opposite changes in the hopping rate, and hence the probability of recombination. For

rods, separation is always perpendicular to the field. Blends and bicontinuous struc-

tures have interfaces at angles from 0 (field-assisted separation) to 180 degrees (field

impaired separation) with respect to the field and so might beexpected to have a similar

efficiency to the rods. However, initially separated charges will have multiple attempts

at geminate recombination if initial separation is againstthe field, as they will need

to explore the interface to find a way out. Thus, the detrimental effect of an interface

where separation is against the field, when compared to perpendicular separation, is far

greater than the advantage of separation with the field.

To confirm this argument, bilayer structures were created for separation at differ-

ent anglesθ to the fieldF . The fraction of charges successfully escaping geminate

recombination,ηgs, were obtained forF = 1 × 107 Vm−1. The results areηgs =

78%,74%,17% forθ = 0, 90 and 180 degrees respectively in agreement with.13 The

magnitude of this effect shows it is the dominant limiting mechanism for efficiency.

Hence, as the interfaces in the blend and bicontinuous morphologies have no preferred

direction with respect to the field, the results can be expected to be noticeably below

those for the rods, whereθ is always 90 degrees, as we find. We do not anticipate this

to be a problem when implementing these structures in DSCs, as the device structure

screens out the internal field.

We can conclude that bicontinuous, triply-periodic minimal surface morphologies

15

(a)

(b)

Figure 3: Recombination fraction of geminate pairsηgr (a) and of nongeminate pairs,bimolecular recombination,ηbr (b) for all morphology classes, as a function of featuresize. Symbols and line types as for 2.

may not enhance polymer blend solar cell efficiency as much ashoped. However, op-

timised bicontinuous structures could be made to have higher efficiencies than blends

with an optimal morphology, and the reproducibility of these structures further en-

hances their practical appeal. Vertical rod structures maybe the best option if they

can be made defect free, as they exhibit a far superior performance, A very narrow

rod structure is the most desirable, to improve the exciton dissociation efficiency. In

order to maintain good charge collection, high mobility materials would be required,

in order to reduce geminate recombination and evacuate charges quickly to avoid sub-

sequent loss. We hope that this paper will stimulate furthermeasurements on cells

based on bicontinuous morphologies, whether by using diblock copolymers or hybrid

16

organic/inorganic cells. Finally, whilst the biggest limitation to solar cell efficiency

is likely to be optical absorption, more highly absorbing polymers are of little use if

their structure and chemical properties are not tailored tohandle the increased charge

density efficiently.

Acknowledgements

This work is supported by the European Union Framework 6 project MODECOM

(NMP-CT-2006-016434). RGEK thanks the UK Engineering and Physical Sciences

Research Council for a studentship. We would like to thank Chris Groves for reading

the manuscript.

Supporting information

Descriptions of how gyroid and diamond triply-periodic minimal surfaces have been

created along with IQEs and the geminate and bimolecular recombination levels at 5

suns illumination. This material is available free of charge via the Internet at http://www.rsc.org.

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